Let $ABC$ be a triangle with incircle $\omega$. Let $A_0,B_0,C_0$ be points outside $\omega$. The tangents from $A_0$ to $\omega$ intersect $BC$ at $A_1,A_2$. Define $B_1,B_2$ and $C_1,C_2$ similarly. Is it true that $A_1,A_2,B_1,B_2,C_1,C_2$ lie on a conic $\Gamma$ if and only if $\Delta ABC$ and $\Delta A_0B_0C_0$ are perspective from a point? I find this to be true in all cases I could possibly check using Geogebra. I suppose it is true. Yet I have no proof or reference on this fact. So something on this topic would be helpful. Thanks a lot.
As noted by Todd this is a problem in projective geometry of conics. So we can use the duality principle and reformulate the "only if" part by the following statement (the other part is equivalent):
Suppose $C$ and $D$ are two conics in the projective plane. For every point $A$ in $C$, we associate a line $T(A)$ in this way: Let $A_1,A_2$ are two other intersection points of tangents of $A$ to $D$ with the conic $C$. Define $T(A)$ to be the line $A_1A_2$. Then for every three points $X,Y,Z$ on $C$, $\Delta XYZ$ is perspective to the triangle with edges $T(X),T(Y), T(Z)$.
To see the equivalence with the original problem, consider $X,Y,Z$ to be the dual points of edges of $\Delta ABC$, $C$ the dual of $\omega$ and $D$ the dual of the conic through $A_1,A_2,B_1,B_2,C_1,C_2$.
(All conics here are nonsingular.)
We use two lemmas:
Lemma 1: Two triangles are perspective iff there exists a polarity which sends every vertex of one of them to the corresponding edge of the other.
Lemma 2: Let $T$ ba a correlation (projective isomorphism of the plane to its dual) and $C$ a conic. If for every points $x,y\in C$, $x \in T(y) \Longleftrightarrow y\in T(x)$, then $T$ is a polarity.
Proof: $T$ is equal to its dual on $C$.
Now for the proof of our statement it suffices to see that $T$ is an algebraic isomorphism between $C$ and dual of a conic in the plane. (The only thing that should be checked is injectivity of $T$ which is true in the generic case. More precisely when there not exists a quadrilateral inscribed in $C$ and circumscribed about $D$.) So $T$ can be extended to a correlation in the plane which obviously satisfies the condition of Lemma 2 for $C$. Therefore $T$ is a restriction of a polarity and Lemma 1 completes the proof.